Studies on the Painleve equations

Painleve 方程的研究

• 批准号: 08454005
• 负责人: UMEMURA Hiroshi
• 金额: $ 3.07万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454005/
• 关键词: Painleve equations; Special polynomials; Baecklund transformations; Painlev'e equations; BaeckLund変換; Painleve方程式; 特殊関数; モノドロミ-保変形

(1) We succeeded in generating special polynomials by Painleve equations, which generalize the Yablonskii-Vorob'ev polynomial for the second Painleve equation and the Okamoto polynomial for the fourth Painleve equation, Polynomials of particular interest appeared in algebraic solutions of the third, the fifth, and the sixth equations.(2) As for the polynomials associated with algebraic solutions of the sixth equation, we formulated a conjecture that each coefficient of the polynomials is equal to the degree of the representation of GL given by an explicit Young diagram.(3) The above conjecture was recently settled.

(1)我们成功地用Painleve方程生成了特殊的多项式，它推广了第二类Painleve方程的Yablonskiii-Vorob 'ev多项式和第四类Painleve方程的Okamoto多项式。特别有趣的多项式出现在第三、第五和第六类方程的代数解中。(2)对于与第六个方程的代数解相关的多项式，我们提出了一个猜想，即多项式的每个系数都等于由显式Young图给出的GL表示的次数。(3)上述猜测最近得到了解决。

项目成果

H.UMEMURA: "Lie-Drach-Vesiof Theory" Adv.Shudies in Pure Math.25. 364-358 (1997)

H.UMEMURA：“Lie-Drach-Vesiof 理论”Adv.Shudies in Pure Math.25。

H.Umemura: "Solutions of the second and fourth Painleve equations" Nagoya Math.J.148. 151-198 (1997)

H.Umemura：“第二和第四 Painleve 方程的解”Nagoya Math.J.148。

H.UMEMURA: "Galoir theory of algebrain and differential equaters" Nagoya Math.J.144. 1-58 (1996)

H.UMEMURA：“代数和微分赤道的伽罗瓦理论”Nagoya Math.J.144。

H.UMEMURA: "Special polynomials asociated noth the Painleve equations" Proc.Workshop on the special functions. (発表予定).

H.UMEMURA：“与 Painleve 方程相关的特殊多项式”Proc.Workshop 关于特殊函数（待介绍）。

H.UMEMURA: "Differential Galuig theory of infinite dimension" Nagoya Math.J.144. 59-135 (1996)

H.UMEMURA：“无限维微分Galuig理论”Nagoya Math.J.144。